Is Rolle's Theorem true when the function..

Question

Is Rolle's Theorem true when the function is not continuous at the end points?

Suppose we define a function $f(x)=x$ for $0<x\leq 1$ and define $f(0)=1$ then it satisfies all the conditions of above theorem, except continuity at $x=0$. Is Rolle's Theorem true in such cases?

Answer 1

We have $f(1)-f(0)=0.$ Suppose that Rolle's theorem is true. Then there is $t \in (0,1)$ such that

$$0=f'(t)=1,$$

which is absurd.

Answer 2

You can redefine function in start/end of interval, so, formally, to have function not continuous, but you need $f(a+)=f(b−)$ to save Rolle's soul.